Let x0 be the point of local maxima of f(x)= veca ⋅( vecb × vecc), where veca=x hati-2 hatj+3 hatk vec b =-2 hat i + x hat j - hat k and vec c =7 hat i -2 hat j + x hat k . Then the value of vec a ⋅ vec b + vec b ⋅ vec c + vec c ⋅ vec a at x = x 0 is :

Question

Let x0 be the point of local maxima of f(x)= veca ⋅( vecb × vecc), where veca=x hati-2 hatj+3 hatk vec b =-2 hat i + x hat j - hat k and vec c =7 hat i -2 hat j + x hat k . Then the value of vec a ⋅ vec b + vec b ⋅ vec c + vec c ⋅ vec a at x = x 0 is : (A) -30 (B) 14 (C) -4 (D) -22

Tardigrade Admin · Accepted Answer

f(x) = veca .( vecb× vecc) = |x&-2&3 -2&x&-1 7&-2&x| = x3 - 27 x + 26 f'( x )=3 x 2-27=0 ⇒ x =± 3 and f''(-3)<0 ⇒ local maxima at x = x 0=-3 Thus, vec a =-3 hat i -2 hat j +3 hat k vec b =-2 hat i -3 hat j - hat k and vec c =7 hat i -2 hat j -3 hat k ⇒ vec a ⋅ vec b + vec b ⋅ vec c + vec c ⋅ vec a =9-5-26=-22

Q. Let x0​ be the point of local maxima of f(x)=a⋅(b×c), where a=xi^−2j^​+3k^ b=−2i^+xj^​−k^ and c=7i^−2j^​+xk^. Then the value of a⋅b+b⋅c+c⋅a at x=x0​ is :

-30

14

-4

-22

f(x)=a.(b×c)=∣∣​x−27​−2x−2​3−1x​∣∣​ =x3−27x+26 f′(x)=3x2−27=0 ⇒x=±3 and f′′(−3)<0 ⇒ local maxima at x=x0​=−3 Thus, a=−3i^−2j^​+3k^ b=−2i^−3j^​−k^ and c=7i^−2j^​−3k^ ⇒a⋅b+b⋅c+c⋅a =9−5−26=−22

Q. Let $x_{0}$ be the point of local maxima of $f (x) = a \cdot (b \times c),$ where $a = x \hat{i} - 2 \hat{j} + 3 \hat{k}$ $b = - 2 \hat{i} + x \hat{j} - \hat{k}$ and $c = 7 \hat{i} - 2 \hat{j} + x \hat{k}$ . Then the value of $a \cdot b + b \cdot c + c \cdot a$ at $x = x_{0}$ is :